Revista Matemática Iberoamericana

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Volume 35, Issue 7, 2019, pp. 2187–2219
DOI: 10.4171/rmi/1117

Published online: 2019-08-20

The Poincaré half-space of a C$^*$-algebra

Esteban Andruchow[1], Gustavo Corach[2] and Lázaro Recht[3]

(1) Universidad Nacional de General Sarmiento, Los Polvorines, Argentina and Instituto Argentino de Matemática, Buenos Aires
(2) Instituto Argentino de Matemáticas and Universidad de Buenos Aires, Argentina
(3) Universidad Simón Bolívar, Caracas, Venezuela

Let $\mathcal{A}$ be a unital C$^*$-algebra. Given a faithful representation $\mathcal{A}\subset\mathcal B(\mathcal{L})$ in a Hilbert space $\mathcal{L}$, the set $G^+\subset\mathcal{A}$ of positive invertible elements can be thought of as the set of inner products in $\mathcal{L}$, related to $\mathcal{A}$, which are equivalent to the original inner product. The set $G^+$ has a rich geometry, it is a homogeneous space of the invertible group $G$ of $\mathcal{A}$, with an invariant Finsler metric. In the present paper we study the tangent bundle $TG^+$ of $G^+$, as a homogeneous Finsler space of a natural group of invertible matrices in $M_2(\mathcal{A})$, identifying $TG^+$ with the Poincaré half-space $\mathcal H$ of $\mathcal{A}$, $$\mathcal H=\{h\in\mathcal{A}: {\rm Im}(h)\ge 0, {\rm Im}(h) \hbox{ invertible}\}.$$ We show that $\mathcal H\simeq TG^+$ has properties similar to those of a space of non-positive constant curvature.

Keywords: Positive invertible operator, inner product

Andruchow Esteban, Corach Gustavo, Recht Lázaro: The Poincaré half-space of a C$^*$-algebra. Rev. Mat. Iberoam. 35 (2019), 2187-2219. doi: 10.4171/rmi/1117